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Chapter 1: Q7.1-31E (page 1)

Consider the dynamical system x→(t+1)=[1.100⅄]X→(t).

Sketch a phase portrait of this system for the given values of λ:

λ=1

Short Answer

Expert verified

The sketch a phase portrait of this system for the given values ofis shown as below:

Step by step solution

01

Definition of matrix

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

02

Sketch the phase portrait

Consider the given matrix,

A=1.1001

Clearly, A is diagonal, so for any:

X0=X1X2

, we have:

AtX0=(1.1)t001X1X2=(1.1)tX1X2

So, the required sketch is shown as below using the technology.

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Most popular questions from this chapter

Question: There exists a 2x2 matrix such thatA[12]=[34].

Let \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\0\\{ - 2}\end{array}} \right],{v_2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\1\\8\end{array}} \right],\) and \({\rm{y = }}\left[ {\begin{array}{*{20}{c}}h\\{ - 5}\\{ - 3}\end{array}} \right]\). For what values(s) of \(h\) is \(y\) in the plane generated by \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\)

Use Theorem 7 in section 1.7 to explain why the columns of the matrix Aare linearly independent.

\(A = \left( {\begin{aligned}{*{20}{c}}1&0&0&0\\2&5&0&0\\3&6&8&0\\4&7&9&{10}\end{aligned}} \right)\)

Give a geometric description of span \(\left\{ {{v_1},{v_2}} \right\}\) for the vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}8\\2\\{ - 6}\end{array}} \right]\) and \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{12}\\3\\{ - 9}\end{array}} \right]\).

In Exercises 11 and 12, determine if \({\rm{b}}\) is a linear combination of \({{\mathop{\rm a}\nolimits} _1},{a_2}\) and \({a_3}\).

12.
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